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Register a new userConsistency of Empirical Bayes And Kernel Flow For Hierarchical Parameter Estimation
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Yifan Chen
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Gaussian process regression has proven very powerful in statistics, machine learning and inverse problems. A crucial aspect of the success of this methodology, in a wide range of applications to complex and real-world problems, is hierarchical modeling and learning of hyperparameters. The purpose of this paper is to study two paradigms of learning hierarchical parameters: one is from the probabilistic Bayesian perspective, in particular, the empirical Bayes approach that has been largely used in Bayesian statistics; the other is from the deterministic and approximation theoretic view, and in particular the kernel flow algorithm that was proposed recently in the machine learning literature. Analysis of their consistency in the large data limit, as well as explicit identification of their implicit bias in parameter learning, are established in this paper for a Matern-like model on the torus. A particular technical challenge we overcome is the learning of the regularity parameter in the Matern-like field, for which consistency results have been very scarce in the spatial statistics literature. Moreover, we conduct extensive numerical experiments beyond the Matern-like model, comparing the two algorithms further. These experiments demonstrate learning of other hierarchical parameters, such as amplitude and lengthscale; they also illustrate the setting of model misspecification in which the kernel flow approach could show superior performance to the more traditional empirical Bayes approach.
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We develop a Nonparametric Empirical Bayes (NEB) framework for compound estimation in the discrete linear exponential family, which includes a wide class of discrete distributions frequently arising from modern big data applications. We propose to directly estimate the Bayes shrinkage factor in the generalized Robbins formula via solving a scalable convex program, which is carefully developed based on a RKHS representation of the Steins discrepancy measure. The new NEB estimation framework is flexible for incorporating various structural constraints into the data driven rule, and provides a unified approach to compound estimation with both regular and scaled squared error losses. We develop theory to show that the class of NEB estimators enjoys strong asymptotic properties. Comprehensive simulation studies as well as analyses of real data examples are carried out to demonstrate the superiority of the NEB estimator over competing methods.
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Although the operator (spectral) norm is one of the most widely used metrics for covariance estimation, comparatively little is known about the fluctuations of error in this norm. To be specific, let $hatSigma$ denote the sample covariance matrix of $n$ observations in $mathbb{R}^p$ that arise from a population matrix $Sigma$, and let $T_n=sqrt{n}|hatSigma-Sigma|_{text{op}}$. In the setting where the eigenvalues of $Sigma$ have a decay profile of the form $lambda_j(Sigma)asymp j^{-2beta}$, we analyze how well the bootstrap can approximate the distribution of $T_n$. Our main result shows that up to factors of $log(n)$, the bootstrap can approximate the distribution of $T_n$ at the dimension-free rate of $n^{-frac{beta-1/2}{6beta+4}}$, with respect to the Kolmogorov metric. Perhaps surprisingly, a result of this type appears to be new even in settings where $p< n$. More generally, we discuss the consequences of this result beyond covariance matrices and show how the bootstrap can be used to estimate the errors of sketching algorithms in randomized numerical linear algebra (RandNLA). An illustration of these ideas is also provided with a climate data example.
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